Units of Wavenumbers

Yet a third spectral unit, commonly used in spectroscopy, is wavenumber, the number of waves per cm: \(\sigma = ν /100c\) cm^-1. Converting (1) to these units gives

\[L_{\sigma}=\left|\frac{d v}{d \sigma}\right| L_{v}=(100 c) L_{v}\]

\[L_{\sigma}=2 \times 10^{8} h c^{2} \sigma^{3} \frac{1}{e^{100 h c \sigma / k T}-1} \quad \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1} \label{m}\tag{13}\]

Again, the peak is where the derivative with respect to wavenumber vanishes:

\[0=\frac{d L_{\sigma}}{d \sigma}=6 \times 10^{8} h c^{2} \sigma^{2} \frac{1}{e^{100 h c \sigma / k T}-1}-2 \times 10^{8} h c^{2} \sigma^{3} \frac{(100 h c / k T) e^{100 h c \sigma / k T}}{\left(e^{100 h c \sigma / k T}-1\right)^{2}}\]

\begin{array}{l} 0=3-\frac{100 \mathrm{hc} \sigma}{\mathrm{kT}} \frac{e^{100 \mathrm{hc} \sigma / \mathrm{kT}}}{e^{100 \mathrm{hc\sigma} / \mathrm{kT}}-1} \quad \text { let } x=\frac{100 \mathrm{hc\sigma}}{\mathrm{kT}} \newline 3\left(1-e^{-\mathrm{x}}\right)=x \newline x=a_{3} \approx 2.82143937212 \end{array}

so

\[\sigma_{\text {peak}}=\frac{a_{3} k T}{100 h c} \mathrm{cm}^{-1} \label{n}\tag{14}\]

The peak value is

\[L_{\sigma, \text { peak }}=2 \times 10^{8} h c^{2}\left(\frac{a_{3} k T}{100 h c}\right)^{3} \frac{1}{e^{100 h c}\left(\frac{a_{3} k T}{100 h c}\right) / k T}-1\]

\[L_{\sigma, \text { peak }}=\frac{200 a_{3}^{3} k^{3}}{h^{2} c} \frac{1}{e^{a_{3}}-1} T^{3} \quad \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1} \label{o}\tag{15}\]

The spectral photon radiance is found by dividing \(L_{\lambda}\) by the energy of a photon, \(100hc \sigma\):

\[L_{\sigma}^{P}=\frac{L_{\sigma}}{100 h c \sigma}=2 \times 10^{6} c \sigma^{2} \frac{1}{e^{100 h c \sigma / k T}-1} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1} \label{p}\tag{16}\]

We next find the wavenumber at the peak of the spectral photon radiance:

\[0=\frac{d L_{\sigma}^{P}}{d \sigma}=4 \times 10^{6} c \sigma \frac{1}{e^{100 h c \sigma / k T}-1}-2 \times 10^{6} c \sigma^{2} \frac{(100 h c / k T) e^{100 h c \sigma / k T}}{\left(e^{100 h c \sigma / k T}-1\right)^{2}}\]

\begin{array}{ll} 0=2-\frac{100 h c \sigma}{k T} \frac{e^{100 h c \sigma} / k T}{e^{100 h c \sigma / k T}-1} & \text { let } x=\frac{100 h c \sigma}{k T} \newline 2\left(1-e^{-x}\right)=x & x=a_{2} \approx 1.59362426004 \end{array}

and

\[\sigma_{\text {peak }}^{P}=\frac{a_{2} k}{100 h c} T \quad \mathrm{cm}^{-1} \label{q}\tag{17}\]

The peak spectral photon radiance is

\[L_{\sigma, \text { peak }}^{p}=2 \times 10^{6} c\left(\frac{a_{2} k T}{100 h c}\right)^{2} \frac{1}{e^{\frac{100 h c\left(a_{2} k T / 100 h c\right)}{k T}-1}}\]

\[L_{\sigma, \text { peak }}^{P}=200 \frac{a_{2}^{2} k^{2}}{h^{2} c} \frac{1}{e^{a_{3}}-1} T^{2} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1} \label{r}\tag{18}\]

Fig. 3 shows plots of \(L_{\sigma}\) and \(L_{\sigma}^{P}\) for various temperatures. Note again the important difference between the spectral radiance and spectral photon radiance.

Fig. 3 -- Spectral radiance, \(L_{\sigma}\) , (top) and the spectral photon radiance, \(L_{\sigma}^{P}\), (bottom) as a function of wavenumber, \(\sigma\), for various temperatures. The small black dots indicate the wavenumber and value of the peak, at 10 K temperature intervals. Note that \(L_{\sigma}\) and \(L_{\sigma}^{P}\) have different wavenumber dependences. Although the peak wavenumber is proportional to \(T\) for both quantities, \(L_{\sigma}\) peaks at a higher wavenumber than \(L_{\sigma}^{P}\). Furthermore, the peak value of \(L_{\sigma}\) increases as \(T^3\), whereas the peak value of \(L_{\sigma}^{P}\) increases as \(T^2\).